PERIODIC SOLUTIONS FOR MCKEAN-VLASOV SDES UNDER PERIODIC DISTRIBUTION-DEPENDENT LYAPUNOV CONDITIONS

The paper proves existence of T‑periodic solutions for McKean–Vlasov SDEs by working with a coupled process on Rd×P(Rd), establishing a periodic probability for its transition kernel under a distribution‑dependent Lyapunov condition (H), and then projecting to obtain a periodic solution to the original MVSDE; the key steps (definition of LV on C^{1,2,(1,1)}, truncation, occupation bounds, Theorem 3.10, and Theorem 4.3) are explicit and coherent . By contrast, the candidate solution’s Krylov–Bogoliubov construction on Zt=(Xt,μt) correctly yields periodic measures on the extended space, but it incorrectly concludes that each time‑slice ρτ must be supported on a single “graph” point δμτ⊗μτ; the consistency identity only forces ρτ to be a μ‑mixture of δμ⊗μ and does not collapse to a singleton without an extra extremality/uniqueness argument. This gap prevents the candidate from deriving a nonlinear Fokker–Planck characterization with a deterministic μτ and hence from completing the existence proof. The paper avoids this pitfall precisely by using the coupled process and the identity P(s,(x,δx),t,·)=LXt×δLXt, which enforces the graph structure needed to project to a periodic MVSDE solution .